Chapter 22 The Electric Field II: Continuous Charge Distributions


 Sheryl Henderson
 1 years ago
 Views:
Transcription
1 Chapte The lectic Field II: Continuous Chage Distibutions 1 [M] A unifom line chage that has a linea chage density l equal to.5 nc/m is on the x axis between x and x 5. m. (a) What is its total chage? Find the electic field on the x axis at (b) x 6. m, (c) x 9. m, and (d) x 5 m. (e) stimate the electic field at x 5 m, using the appoximation that the chage is a point chage on the x axis at x.5 m, and compae you esult with the esult calculated in Pat (d). (To do this you will need to assume that the values given in this poblem statement ae valid to moe than two significant figues.) Is you appoximate esult geate o smalle than the exact esult? xplain you answe. Pictue the Poblem We can use the definition of λ to find the total chage of the line of chage and the expession fo the electic field on the axis of a finite line of chage to evaluate x at the given locations along the x axis. In Pat (d) we can apply Coulomb s law fo the electic field due to a point chage to appoximate the electic field at x 5 m. (a) Use the definition of linea chage density to expess in tems of λ: λl 18nC (.5nC/m)( 5.m) 17.5nC xpess the electic field on the axis of a finite line chage: x ( x ) x k ( x L) (b) ubstitute numeical values and evaluate x at x 6. m: x 9 ( ) ( N m /C )( 17.5nC) 6.m ( 6.m)( 6.m 5.m) 6 N/C (c) ubstitute numeical values and evaluate x at x 9. m: x 9 ( ) ( N m /C )( 17.5nC) 9.m ( 9.m)( 9.m 5.m) 4.4 N/C (d) ubstitute numeical values and evaluate x at x 5 m: x 9 ( ) ( N m /C )( 17.5nC) 5m ( 5m)( 5m 5.m).568 mn/c.6mn/c 89
2 9 Chapte (e) Use Coulomb s law fo the electic field due to a point chage to obtain: k x ( x) x
3 The lectic Field II: Continuous Chage Distibutions 91 ubstitute numeical values and evaluate x (5 m): x 9 ( ) ( N m /C )( 17.5nC) 5m ( 5m.5 m).56774mn/c.6 mn/c This esult is about.1% less than the exact value obtained in (d). This suggests that the line of chage is too long fo its field at a distance of 5 m to be modeled exactly as that due to a point chage. 17 [M] A ing that has adius a lies in the z plane with its cente at the oigin. The ing is unifomly chaged and has a total chage. Find z on the z axis at (a) z.a, (b) z.5a, (c) z.7a, (d) z a, and (e) z a. (f) Use you esults to plot z vesus z fo both positive and negative values of z. (Assume that these distances ae exact.) z Pictue the Poblem We can use z πkq 1 to find the electic z + a field at the given distances fom the cente of the chaged ing. (a) valuate z (.a): (.a) (b) valuate z (.5a): (.5a) (c) valuate z (.7a): (.7a) z z z k(.a) (.a) + a [ ] k.189 a k(.5a) (.5a) + a [ ] k.58 a k(.7a) (.7a) + a [ ] k.85 a (d) valuate z (a): ka z ( a) k.54 [ a + a ] a
4 9 Chapte (e) valuate z (a): ka z ( a) k.179 [( a) + a ] a
5 The lectic Field II: Continuous Chage Distibutions 9 (f) The field along the x axis is plotted below. The z coodinates ae in units of z/a and is in units of k/a..4. x z/a 18 A nonconducting disk of adius a lies in the z plane with its cente at the oigin. The disk is unifomly chaged and has a total chage. Find z on the z axis at (a) z.a, (b) z.5a, (c) z.7a, (d) z a, and (e) z a. (f) Use you esults to plot z vesus z fo both positive and negative values of z. (Assume that these distances ae exact.) z Pictue the Poblem We can use z π kq 1, whee a is the adius z + a of the disk, to find the electic field on the axis of a chaged disk. The electic field on the axis of a chaged disk of adius a is given by: z πk 1 1 z z z + a z + a (a) valuate z (.a): z (.a) 1.4.a (.a) + a
6 94 Chapte (b) valuate z (.5a): z (.5a) a (.5a) + a (c) valuate z (.7a): z (.7a) 1.1.7a (.7a) + a (d) valuate z (a): z ( a) a a + a (e) valuate z (a): z ( a) 1.58 a ( a) + a The field along the x axis is plotted below. The x coodinates ae in units of z/a and is in units of x z/a
7 The lectic Field II: Continuous Chage Distibutions 95 7 A squae that has 1cmlong edges is centeed on the x axis in a ρ egion whee thee exists a unifom electic field given by (. kn/c)iˆ. (a) What is the electic flux of this electic field though the suface of a squae if the nomal to the suface is in the +x diection? (b) What is the electic flux though the same squae suface if the nomal to the suface makes a 6º angle with the y axis and an angle of 9 with the z axis? Pictue the Poblem The definition of electic flux isφ ρ nˆda. We can apply this definition to find the electic flux though the squae in its two oientations. (a) Apply the definition of φ to find the flux of the field when the squae is paallel to the yz plane: φ (.kn/c) (.kn/c) (.kn/c)(.1m). N m iˆ ida ˆ da /C φ (. kn/c) i : (. kn/c) (b) Poceed as in (a) with ˆ n ˆ cos (. kn/c)(.1 m) 17 N m /C cos da cos da cos ρ 9 [M] An electic field is given by sign( x) ( N/C)iˆ, whee sign(x) equals 1 if x <, if x, and +1 if x >. A cylinde of length cm and adius 4. cm has its cente at the oigin and its axis along the x axis such that one end is at x +1 cm and the othe is at x 1 cm. (a) What is the electic flux though each end? (b) What is the electic flux though the cuved suface of the cylinde? (c) What is the electic flux though the entie closed suface? (d) What is the net chage the cylinde? Pictue the Poblem The field at both cicula faces of the cylinde is paallel to the outwad vecto nomal to the suface, so the flux is just A. Thee is no flux though the cuved suface because the nomal to that suface is pependicula to ρ. The net flux though the closed suface is elated to the net chage by Gauss s law.
8 96 Chapte (a) Use Gauss s law to calculate the flux though the ight cicula suface: Apply Gauss s law to the left cicula suface: φ φ ight left ρ ρ ight nˆ ight ( N/C) iˆ iˆ ( π )(.4 m) 1.5 N m left nˆ left A A /C ( N/C) iˆ ( iˆ )( π )(.4 m) 1.5 N m /C (b) Because the field lines ae paallel to the cuved suface of the cylinde: φ cuved (c) xpess and evaluate the net flux though the entie cylindical suface: φ net φ + φ + φ ight left 1.5 N m /C N m /C +. N m /C cuved (d) Apply Gauss s law to obtain: φ net 4πk φnet 4πk ubstitute numeical values and evaluate : 4 π 9 ( N m /C ).7 1. N m 11 C /C
9 The lectic Field II: Continuous Chage Distibutions 97 1 A point chage (q +. μc) is at the cente of an imaginay sphee that has a adius equal to.5 m. (a) Find the suface aea of the sphee. (b) Find the magnitude of the electic field at all points on the suface of the sphee. (c) What is the flux of the electic field though the suface of the sphee? (d) Would you answe to Pat (c) change if the point chage wee moved so that it was the sphee but not at its cente? (e) What is the flux of the electic field though the suface of an imaginay cube that has 1.mlong edges and encloses the sphee? Pictue the Poblem We can apply Gauss s law to find the flux of the electic field though the suface of the sphee. (a) Use the fomula fo the suface aea of a sphee to obtain: A 4π 4π.14m (.5 m).14 m (b) Apply Coulomb s law to find : 4π 1 q 4π N/C 1.μC 1 ( C /N m )(.5 m) N/C (c) Apply Gauss s law to obtain: nda ˆ da 4 ( N/C)(.14 m ) φ ρ N m /C (d) No. The flux though the suface is independent of whee the chage is located the sphee. (e) Because the cube encloses the sphee, the flux though the suface of the sphee will also be the flux though the cube: φ cube N m /C 4 Because the fomulas fo Newton s law of gavity and fo Coulomb s law have the same invesesquae dependence on distance, a fomula analogous to the fomula fo Gauss s law can be found fo gavity. The gavitational field g ρ at a location is the foce pe unit mass on a test mass m placed at that location. Then, fo a point mass m at the oigin, the gavitational field g ρ at some position
10 98 Chapte ( ρ ) is g ρ ( Gm )ˆ. Compute the flux of the gavitational field though a spheical suface of adius R centeed at the oigin, and veify that the gavitational analog of Gauss s law is φ net 4πGm. Pictue the Poblem We ll define the flux of the gavitational field in a manne that is analogous to the definition of the flux of the electic field and then substitute fo the gavitational field and evaluate the integal ove the closed spheical suface. Define the gavitational flux as: φ g ρ nˆda g ubstitute fo g ρ and evaluate the integal to obtain: φ net Gm Gm Gm 4πGm ˆ nˆ da da ( 4π ) 4 Conside the solid conducting sphee and the concentic conducting spheical shell in Figue 41. The spheical shell has a chage 7. The solid sphee has a chage +. (a) How much chage is on the oute suface and how much chage is on the inne suface of the spheical shell? (b) uppose a metal wie is now connected between the solid sphee and the shell. Afte electostatic equilibium is eestablished, how much chage is on the solid sphee and on each suface of the spheical shell? Does the electic field at the suface of the solid sphee change when the wie is connected? If so, in what way? (c) uppose we etun to the conditions in Pat (a), with + on the solid sphee and 7 on the spheical shell. We next connect the solid sphee to gound with a metal wie, and then disconnect it. Then how much total chage is on the solid sphee and on each suface of the spheical shell? Detemine the Concept The chages on a conducting sphee, in esponse to the epulsive Coulomb foces each expeiences, will sepaate until electostatic equilibium conditions exit. The use of a wie to connect the two sphees o to gound the oute sphee will cause additional edistibution of chage. (a) Because the oute sphee is conducting, the field in the thin shell must vanish. Theefoe,, unifomly distibuted, esides on the inne suface, and 5, unifomly distibuted, esides on the oute suface.
11 The lectic Field II: Continuous Chage Distibutions 99
12 1 Chapte (b) Now thee is no chage on the inne suface and 5 on the oute suface of the spheical shell. The electic field just outside the suface of the inne sphee changes fom a finite value to zeo. (c) In this case, the 5 is dained off, leaving no chage on the oute suface and on the inne suface. The total chage on the oute sphee is then. 41 A nonconducting solid sphee of adius 1. cm has a unifom volume chage density. The magnitude of the electic field at. cm fom the sphee s cente is N/C. (a) What is the sphee s volume chage density? (b) Find the magnitude of the electic field at a distance of 5. cm fom the sphee s cente. Pictue the Poblem (a) We can use the definition of volume chage density, in conjunction with quation 18a, to find the sphee s volume chage density. (b) We can use quation 18b, in conjunction with ou esult fom Pat (a), to find the electic field at a distance of 5. cm fom the solid sphee s cente. (a) The solid sphee s volume chage density is the atio of its chage to its volume: (1) V ρ 4 πr Fo R, quation 18a gives the electic field at a distance fom the cente of the sphee: 1 () 4π olving fo yields: 4π ubstitute fo in equation (1) and simplify to obtain: 4π ρ 4 πr R ubstitute numeical values and evaluate ρ: 1 ( 1 C /N m )( N/C)(. cm) ρ. μc/m ( 1. cm) 1.997μC/m (b) Fo R, the electic field at a distance fom the cente of the sphee is given by: 1 () 4π R
13 The lectic Field II: Continuous Chage Distibutions 11 xpess fo R: 4 ρv π ρ sphee adius is whose ubstituting fo in equation () and simplifying yields: 1 4π 4 π R 4 ρ ρ R ubstitute numeical values and evaluate (5. cm): 4 ( ) ( 1.997μC/m )( 5. cm). cm 1 ( C /N m )( 1. cm) 5 47 N/C 4 [M] A sphee of adius R has volume chage density ρ B/ fo < R, whee B is a constant and ρ fo > R. (a) Find the total chage on the sphee. (b) Find the expessions fo the electic field and outside the chage distibution (c) ketch the magnitude of the electic field as a function of the distance fom the sphee s cente. Pictue the Poblem We can find the total chage on the sphee by expessing the chage dq in a spheical shell and integating this expession between and R. By symmety, the electic fields must be adial. To find the chaged sphee we choose a spheical Gaussian suface of adius < R. To find outside the chaged sphee we choose a spheical Gaussian suface of adius > R. On each of these sufaces, is constant. Gauss s law then elates to the total chage the suface. (a) xpess the chage dq in a shell of thickness d and volume 4π d: Integate this expession fom to R to find the total chage on the sphee: dq 4π ρd 4π 4πBd R 4πB d πbr B [ πb ] d R (b) Apply Gauss s law to a spheical suface of adius > R that is concentic with the nonconducting sphee to obtain: 1 da o 4π
14 1 Chapte olving fo yields: ( > R) 4π 1 k kπbr BR Apply Gauss s law to a spheical suface of adius < R that is concentic with the nonconducting sphee to obtain: 1 da 4π olving fo yields: ( < R) 4π B πb 4π
15 The lectic Field II: Continuous Chage Distibutions 1 (c) The following gaph of vesus /R, with in units of B/( ), was plotted using a speadsheet pogam /R Remaks: Note that ou esults fo (a) and (b) agee at R. 46 Fo you senio poject you ae in chage of designing a Geige tube fo detecting adiation in the nuclea physics laboatoy. This instument will consist of a long metal cylindical tube that has a long staight metal wie unning down its cental axis. The diamete of the wie is to be.5 mm and the diamete of the tube will be 4. cm. The tube is to be filled with a dilute gas in which electical dischage (beakdown) occus when the electic field eaches N/C. Detemine the maximum linea chage density on the wie if beakdown of the gas is not to happen. Assume that the tube and the wie ae infinitely long. Pictue the Poblem The electic field of a line chage of infinite length is given 1 λ by, whee is the distance fom the cente of the line of chage and π λ is the linea chage density of the wie. The electic field of a line chage of infinite length is given by: 1 λ π Because vaies invesely with, its maximum value occus at the suface of the wie whee R, the adius of the wie: max 1 λ R π olving fo λ yields: λ π R max ubstitute numeical values and evaluate λ:
16 14 Chapte λ π C N m N C 6 (.5 mm) nc/m 48 how that the electic field due to an infinitely long, unifomly chaged thin cylindical shell of adius a having a suface chage density is given by the following expessions: fo R < a and R a R ( ) fo R > a. Pictue the Poblem Fom symmety, the field in the tangential diection must vanish. We can constuct a Gaussian suface in the shape of a cylinde of adius and length L and apply Gauss s law to find the electic field as a function of the distance fom the centeline of the infinitely long, unifomly chaged cylindical shell. Apply Gauss s law to the cylindical suface of adius and length L that is concentic with the infinitely long, unifomly chaged cylindical shell: 1 n da o πl R whee we ve neglected the end aeas because no thee is no flux though them. olve fo R : R πl k L Fo < R, and: ( < R) Fo > R, λl and: ( > R) R R kλl kλ k L R ( πr ) 49 A thin cylindical shell of length m and adius 6. cm has a unifom suface chage density of 9. nc/m. (a) What is the total chage on the shell? Find the electic field at the following adial distances fom the long axis of the cylinde. (b). cm, (c) 5.9 cm, (d) 6.1 cm, and (e) 1. cm. (Use the esults of Poblem 48.) Pictue the Poblem We can use the definition of suface chage density to find the total chage on the shell. Fom symmety, the electic field in the tangential
17 The lectic Field II: Continuous Chage Distibutions 15 diection must vanish. We can constuct a Gaussian suface in the shape of a cylinde of adius and length L and apply Gauss s law to find the electic field as a function of the distance fom the centeline of the unifomly chaged cylindical shell. (a) Using its definition, elate the suface chage density to the total chage on the shell: ubstitute numeical values and evaluate : A πrl π (.6 m)( m)( 9. nc/m ) 679nC (b) Fom Poblem 48 we have, fo. cm: (c) Fom Poblem 48 we have, fo 5.9 cm: (.cm) ( 5.9cm) (d) Fom Poblem 48 we have, fo 6.1 cm: and () R ( ) ( 6 9. nc/m )(.6 m).1cm 1 ( C /N m )(.61 m) 1. kn/c
18 16 Chapte (e) Fom Poblem 48 we have, fo 1. cm: ( ) ( 1 9. nc/m )(.6 m).cm 1 ( C /N m )(.1 m) 61 N/C 5 An infinitely long nonconducting solid cylinde of adius a has a unifom volume chage density of ρ. how that the electic field is given by the following expessions: R ρ R ( ) fo R < a and R ρ a ( R) fo R > a, whee R is the distance fom the long axis of the cylinde. Pictue the Poblem Fom symmety, the field tangent to the suface of the cylinde must vanish. We can constuct a Gaussian suface in the shape of a cylinde of adius and length L and apply Gauss s law to find the electic field as a function of the distance fom the centeline of the infinitely long nonconducting cylinde. Apply Gauss s law to a cylindical suface of adius and length L that is concentic with the infinitely long nonconducting cylinde: 1 n da o πl R whee we ve neglected the end aeas because thee is no flux though them. olving fo R yields: R πl k L xpess fo < R: ρ ( ) V ρ( π L) ubstitute to obtain: ( πρ L ) k ρ R ( < R) L o, because λ ρπr, λ π R ( < R) R xpess fo > R: ρ ( ) V ρ( πr L)
19 The lectic Field II: Continuous Chage Distibutions 17 ubstitute fo to obtain: ( πρ LR ) k ρr R ( > R) L o, because λ ρπr ( > R) λ π R 5 Figue 4 shows a potion of an infinitely long, concentic cable in coss section. The inne conducto has a chage of 6. nc/m and the oute conducto has no net chage. (a) Find the electic field fo all values of R, whee R is the pependicula distance fom the common axis of the cylindical system. (b) What ae the suface chage densities on the and the outside sufaces of the oute conducto? Pictue the Poblem The electic field is diected adially outwad. We can constuct a Gaussian suface in the shape of a cylinde of adius and length L and apply Gauss s law to find the electic field as a function of the distance fom the centeline of the infinitely long, unifomly chaged cylindical shell. (a) Apply Gauss s law to a cylindical suface of adius and length L that is concentic with the inne conducto: 1 n da πl R whee we ve neglected the end aeas because thee is no flux though them. olving fo R yields: R k (1) L Fo < 1.5 cm, and: R ( <1.5cm) Letting R 1.5 cm, expess fo 1.5 cm < < 4.5 cm: λl πrl ubstitute in equation (1) to obtain: k R ( 1.5cm < < 4.5cm) L kλ ubstitute numeical values and evaluate n (1.5 cm < < 4.5 cm): R ( λl) ( ) ( ) ( 6. nc/m ) ( 18N m/c 1.5cm 4.5cm N m /C ) 9 < <
20 18 Chapte xpess fo 4.5 cm < < 6.5 cm: Letting epesent the chage density on the oute suface, expess fo > 6.5 cm: ubstitute in equation (1) to obtain: and R ( 4.5cm < < 6.5cm) A π RL whee R 6.5 cm. ( π R L) k R R ( > R ) L In (b) we show that 1. nc/m. ubstitute numeical values to obtain: R ( ) ( 1. nc/m )( 6.5cm).5cm > N m/c 1 ( C / N m )
21 The lectic Field II: Continuous Chage Distibutions 19 (b) The suface chage densities on the and the outside sufaces of the oute conducto ae given by: λ πr and λ outside πr outside ubstitute numeical values and 6. nc/m π evaluate and outside : (.45 m) 1. nc/m and 6. nc/m outside π (.65 m) 14.7 nc/m 1. nc/m 59 A thin metal slab has a net chage of zeo and has squae faces that have 1cmlong sides. It is in a egion that has a unifom electic field that is pependicula to its faces. The total chage induced on one of the faces is 1. nc. What is the magnitude of the electic field? Pictue the Poblem Because the metal slab is in an extenal electic field, it will have chages of opposite signs induced on its faces. The induced chage is elated to the electic field by. / Relate the magnitude of the electic field to the chage density on the metal slab: Use its definition to expess : ubstitute fo to obtain: A L L ubstitute numeical values and 1 evaluate : (.1m) ( C /N m ) 9.4 kn/c 1.nC
22 11 Chapte 61 A conducting spheical shell that has zeo net chage has an inne adius R 1 and an oute adius R. A positive point chage q is placed at the cente of the shell. (a) Use Gauss s law and the popeties of conductos in electostatic equilibium to find the electic field in the thee egions: < R 1, R 1 < < R, and > R, whee is the distance fom the cente. (b) Daw the electic field lines in all thee egions. (c) Find the chage density on the inne suface ( R 1 ) and on the oute suface ( R ) of the shell. Pictue the Poblem We can constuct a Gaussian suface in the shape of a sphee of adius with the same cente as the shell and apply Gauss s law to find the electic field as a function of the distance fom this point. The inne and oute sufaces of the shell will have chages induced on them by the chage q at the cente of the shell. (a) Apply Gauss s law to a spheical suface of adius that is concentic with the point chage: olving fo Fo < R 1, yields: q. ubstitute in equation (1) and simplify to obtain: Because the spheical shell is a conducto, a chage q will be induced on its inne suface. Hence, fo R 1 < < R : Fo > R, q. ubstitute in equation (1) and simplify to obtain: 1 n da 4π (1) 4π q kq ( < R1 ) 4π and ( R < < R ) 1 q kq ( > R ) 4π
23 (b) The electic field lines ae shown in the diagam to the ight: The lectic Field II: Continuous Chage Distibutions 111 (c) A chage q is induced on the inne suface. Use the definition of suface chage density to obtain: inne q 4πR 1 A chage q is induced on the oute suface. Use the definition of suface chage density to obtain: oute q 4πR 6 The electic field just above the suface of ath has been measued to typically be 15 N/C pointing downwad. (a) What is the sign of the net chage on ath s suface unde typical conditions? (b)what is the total chage on ath s suface implied by this measuement? Pictue the Poblem We can constuct a spheical Gaussian suface at the suface of ath (we ll assume ath is a sphee) and apply Gauss s law to elate the electic field to its total chage. (a) Because the diection of an electic field is the diection of the foce acting on a positively chaged object, the net chage on ath s suface must be negative. (b)apply Gauss s law to a spheical suface of adius R that is concentic with ath: 1 n da 4πR n olve fo to obtain: ath ath 4 π Rn R k n ubstitute numeical values and evaluate : ath ath 6 ( m) ( 15 N/C) kc 9 N m /C
24 11 Chapte 6 [M] A positive point chage of.5 μc is at the cente of a conducting spheical shell that has a net chage of zeo, an inne adius equal to 6 cm, and an oute adius equal to 9 cm. (a) Find the chage densities on the inne and oute sufaces of the shell and the total chage on each suface. (b) Find the electic field eveywhee. (c) Repeat Pat (a) and Pat (b) with a net chage of +.5 μc placed on the shell. Pictue the Poblem Let the inne and oute adii of the unchaged spheical conducting shell be R 1 and R and q epesent the positive point chage at the cente of the shell. The positive point chage at the cente will induce a negative chage on the inne suface of the shell and, because the shell is unchaged, an equal positive chage will be induced on its oute suface. To solve Pat (b), we can constuct a Gaussian suface in the shape of a sphee of adius with the same cente as the shell and apply Gauss s law to find the electic field as a function of the distance fom this point. In Pat (c) we can use a simila stategy with the additional chage placed on the shell. (a) xpess the chage density on the inne suface: xpess the elationship between the positive point chage q and the chage induced on the inne suface q : inne q inne inne A q + q inne q q inne ubstitute fo qinne and A to obtain: inne q 4πR 1 ubstitute numeical values and.5μc π inne evaluate inne : 4 (.6m).55 μc/m xpess the chage density on the oute suface: Because the spheical shell is unchaged: q oute oute A q oute + qinne ubstitute fo q oute to obtain: oute q 4πR inne ubstitute numeical values and.5μc π oute evaluate oute : 4 (.9m).5 μc/m
25 The lectic Field II: Continuous Chage Distibutions 11 (b) Apply Gauss s law to a spheical suface of adius that is concentic with the point chage: olve fo : 1 n da 4π (1) 4π Fo < R 1 6 cm, q. ubstitute in equation (1) and evaluate ( < 6 cm) to obtain: 9 q kq ( ) ( N m /C )(.5μC) < 6 cm 4π 4 1 (. 1 N m /C)
26 114 Chapte Because the spheical shell is a conducto, a chage q will be induced on its inne suface. Hence, fo 6 cm < < 9 cm: and ( 6 cm < < 9cm) Fo > 9 cm, the net chage the Gaussian suface is q and: kq 1 4 ( > 9cm) (. 1 N m /C) (c) Because in the conducto: q inne and inne.5μc.55 μc/m as befoe. xpess the elationship between the chages on the inne and oute sufaces of the spheical shell: q oute and q oute + q inne.5μc.5μ C  q 6.μC inne oute is now given by: oute 6.μC 4 π (.9m).59 μc/m Fo < R 1 6 cm, q and ( < 6 cm) is as it was in (a): Because the spheical shell is a conducto, a chage q will be induced on its inne suface. Hence, fo 6 cm < < 9 cm: ( < 6 cm) (. 1 N m /C) and ( 6 cm < < 9cm) 4 1 Fo >.9 m, the net chage the Gaussian suface is 6. μc and: kq 9 4 ( > 9cm) ( N m /C )( 6.μ C) ( N m /C) Conside the concentic metal sphee and spheical shells that ae shown in Figue 4. The innemost is a solid sphee that has a adius R 1. A spheical shell suounds the sphee and has an inne adius R and an oute adius R. The sphee and the shell ae both suounded by a second spheical shell that
27 The lectic Field II: Continuous Chage Distibutions 115 has an inne adius R 4 and an oute adius R 5. None of these thee objects initially have a net chage. Then, a negative chage is placed on the inne sphee and a positive chage + is placed on the outemost shell. (a) Afte the chages have eached equilibium, what will be the diection of the electic field between the inne sphee and the middle shell? (b) What will be the chage on the inne suface of the middle shell? (c) What will be the chage on the oute suface of the middle shell? (d) What will be the chage on the inne suface of the outemost shell? (e) What will be the chage on the oute suface of the outemost shell? (f) Plot as a function of fo all values of. Detemine the Concept We can detemine the diection of the electic field between sphees I and II by imagining a test chage placed between the sphees and detemining the diection of the foce acting on it. We can detemine the amount and sign of the chage on each sphee by ealizing that the chage on a given suface induces a chage of the same magnitude but opposite sign on the next suface of lage adius. (a) The chage placed on sphee III has no beaing on the electic field between sphees I and II. The field in this egion will be in the diection of the foce exeted on a test chage placed between the sphees. Because the chage at the cente is negative, the field will point towad the cente. (b) The chage on sphee I ( ) will induce a chage of the same magnitude but opposite sign on sphee II: + (c) The induction of chage + on the inne suface of sphee II will leave its oute suface with a chage of the same magnitude but opposite sign: (d) The pesence of chage on the oute suface of sphee II will induce a chage of the same magnitude but opposite sign on the inne suface of sphee III: + (e) The pesence of chage + on the inne suface of sphee III will leave the oute suface of sphee III neutal:
28 116 Chapte (f) A gaph of as a function of is shown to the ight:
Gauss Law. Physics 231 Lecture 21
Gauss Law Physics 31 Lectue 1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing
More informationGauss s law relates to total electric flux through a closed surface to the total enclosed charge.
Chapte : Gauss s Law Gauss s Law is an altenative fomulation of the elation between an electic field and the souces of that field in tems of electic flu. lectic Flu Φ though an aea ~ Numbe of Field Lines
More informationProblems on Force Exerted by a Magnetic Fields from Ch 26 T&M
Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuentcaying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to
More information2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,
3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects
More informationChapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43
Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.
More informationChapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.
Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming
More informationPhysics 235 Chapter 5. Chapter 5 Gravitation
Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus
More information14. Gravitation Universal Law of Gravitation (Newton):
14. Gavitation 1 Univesal Law of Gavitation (ewton): The attactive foce between two paticles: F = G m 1m 2 2 whee G = 6.67 10 11 m 2 / kg 2 is the univesal gavitational constant. F m 2 m 1 F Paticle #1
More informationFXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.
Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing
More informationVoltage ( = Electric Potential )
V1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage
More information2  ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1
 ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation
More informationRevision Guide for Chapter 11
Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationChapter 2. Electrostatics
Chapte. Electostatics.. The Electostatic Field To calculate the foce exeted by some electic chages,,, 3,... (the souce chages) on anothe chage Q (the test chage) we can use the pinciple of supeposition.
More informationLesson 7 Gauss s Law and Electric Fields
Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual
More informationChapter 4. Electric Potential
Chapte 4 Electic Potential 4.1 Potential and Potential Enegy... 43 4.2 Electic Potential in a Unifom Field... 47 4.3 Electic Potential due to Point Chages... 48 4.3.1 Potential Enegy in a System of
More informationGravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2
F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,
More informationAlgebra and Trig. I. A point is a location or position that has no size or dimension.
Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite
More informationA r. (Can you see that this just gives the formula we had above?)
241 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down  you can pedict (o contol) motion
More informationNotes on Electric Fields of Continuous Charge Distributions
Notes on Electic Fields of Continuous Chage Distibutions Fo discete pointlike electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution
More informationVector Calculus: Are you ready? Vectors in 2D and 3D Space: Review
Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.7. find the vecto defined
More informationNewton s Shell Theorem
Newton Shell Theoem Abtact One of the pincipal eaon Iaac Newton wa motivated to invent the Calculu wa to how that in applying hi Law of Univeal Gavitation to pheicallyymmetic maive bodie (like planet,
More informationMagnetic Field and Magnetic Forces. Young and Freedman Chapter 27
Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew  electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field
More informationUNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Approximate time two 100minute sessions
Name St.No.  Date(YY/MM/DD) / / Section Goup# UNIT 21: ELECTRICAL AND GRAVITATIONAL POTENTIAL Appoximate time two 100minute sessions OBJECTIVES I began to think of gavity extending to the ob of the moon,
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationForces & Magnetic Dipoles. r r τ = μ B r
Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent
More informationThe Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = W/q 0 1V [Volt] =1 Nm/C
Geneal Physics  PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit
More informationGravitation. AP Physics C
Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What
More informationVoltage ( = Electric Potential )
V1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationChapter 30: Magnetic Fields Due to Currents
d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.11 T) i to ue a lage cuent flowing though a wie.
More informationProblem Set 6: Solutions
UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 164 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente
More informationPHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013
PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,
More information81 Newton s Law of Universal Gravitation
81 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,
More informationDeflection of Electrons by Electric and Magnetic Fields
Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An
More informationPY1052 Problem Set 8 Autumn 2004 Solutions
PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ighthand end. If H 6.0 m and h 2.0 m, what
More informationAP Physics Electromagnetic Wrap Up
AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle
More informationExam 3: Equation Summary
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P
More information12. Rolling, Torque, and Angular Momentum
12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationIntroduction to Electric Potential
Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic
More informationPoynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions
Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallelplate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing
More informationPHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013
PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationChapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6
Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe
More informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationCHAPTER 5 GRAVITATIONAL FIELD AND POTENTIAL
CHATER 5 GRAVITATIONAL FIELD AND OTENTIAL 5. Intoduction. This chapte deals with the calculation of gavitational fields and potentials in the vicinity of vaious shapes and sizes of massive bodies. The
More informationIntroduction to Fluid Mechanics
Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body
More informationCoordinate Systems L. M. Kalnins, March 2009
Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean
More informationCharges, Coulomb s Law, and Electric Fields
Q&E 1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded
More informationLecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3
Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each
More informationGAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS ` E MISN0133. CHARGE DISTRIBUTIONS by Peter Signell, Michigan State University
MISN0133 GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS GAUSS S LAW APPLIED TO CYLINDRICAL AND PLANAR CHARGE DISTRIBUTIONS by Pete Signell, Michigan State Univesity 1. Intoduction..............................................
More informationEXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD
260 161. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and
More informationHour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and
Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon
More informationCarterPenrose diagrams and black holes
CatePenose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example
More informationF G r. Don't confuse G with g: "Big G" and "little g" are totally different things.
G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just
More informationMagnetic Field in a TimeDependent Capacitor
Magnetic Field in a TimeDependent Capacito 1 Poblem Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 8544 (Octobe 3, 23) Reconside the classic example of the use of Maxwell s displacement
More informationLab M4: The Torsional Pendulum and Moment of Inertia
M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disklike mass suspended fom a thin od o wie. When the mass is twisted about the
More information(a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of
Homewok VI Ch. 7  Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the
More informationCHAPTER 10 Aggregate Demand I
CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income
More informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationChapter 3 Savings, Present Value and Ricardian Equivalence
Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,
More informationSELFINDUCTANCE AND INDUCTORS
MISN0144 SELFINDUCTANCE AND INDUCTORS SELFINDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. SelfInductance L.........................................
More informationELECTRIC CHARGES AND FIELDS
Chapte One ELECTRIC CHARGES AND FIELDS 1.1 INTRODUCTION All of us have the expeience of seeing a spak o heaing a cackle when we take off ou synthetic clothes o sweate, paticulaly in dy weathe. This is
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationCh. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth
Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationrotation  Conservation of mechanical energy for rotation  Angular momentum  Conservation of angular momentum
Final Exam Duing class (13:55 pm) on 6/7, Mon Room: 41 FMH (classoom) Bing scientific calculatos No smat phone calculatos l ae allowed. Exam coves eveything leaned in this couse. Review session: Thusday
More informationExperiment MF Magnetic Force
Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuentcaying conducto is basic to evey electic moto  tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating
More informationElectrostatic properties of conductors and dielectrics
Unit Electostatic popeties of conductos and dielectics. Intoduction. Dielectic beaking. onducto in electostatic equilibium..3 Gound connection.4 Phenomena of electostatic influence. Electostatic shields.5
More information4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first nonzero digit to
. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate
More informationNuclear models: FermiGas Model Shell Model
Lectue Nuclea mode: FemiGas Model Shell Model WS/: Intoduction to Nuclea and Paticle Physics The basic concept of the Femigas model The theoetical concept of a Femigas may be applied fo systems of weakly
More information92.131 Calculus 1 Optimization Problems
9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle
More information9.5 Volume of Pyramids
Page of 7 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using
More informationTheory and measurement
Gavity: Theoy and measuement Reading: Today: p11  Theoy of gavity Use two of Newton s laws: 1) Univesal law of gavitation: ) Second law of motion: Gm1m F = F = mg We can combine them to obtain the gavitational
More informationLesson 8 Ampère s Law and Differential Operators
Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationStructure and evolution of circumstellar disks during the early phase of accretion from a parent cloud
Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The
More informationMultiple choice questions [70 points]
Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions
More informationA couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance, d. F A F B (= F A
5 Moment of a Couple Ref: Hibbele 4.6, edfod & Fowle: Statics 4.4 couple is a pai of foces, equal in magnitude, oppositely diected, and displaced by pependicula distance, d. d (=  ) Since the foces ae
More informationUNIT CIRCLE TRIGONOMETRY
UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + =   
More informationNUCLEAR MAGNETIC RESONANCE
19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic
More informationSolutions for Physics 1301 Course Review (Problems 10 through 18)
Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal
More informationChapter 2 Coulomb s Law
Chapte Coulomb s Law.1 lectic Chage...3. Coulomb's Law...3 Animation.1: Van de Gaaff Geneato...4.3 Pinciple of Supeposition...5 xample.1: Thee Chages...5.4 lectic Field...7 Animation.: lectic Field
More informationPhysics HSC Course Stage 6. Space. Part 1: Earth s gravitational field
Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe
More informationChapter 17 The Kepler Problem: Planetary Mechanics and the Bohr Atom
Chapte 7 The Keple Poblem: Planetay Mechanics and the Boh Atom Keple s Laws: Each planet moves in an ellipse with the sun at one focus. The adius vecto fom the sun to a planet sweeps out equal aeas in
More informationProblems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)
Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems
More information7 Circular Motion. 71 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary
7 Cicula Motion 71 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o
More informationThe Role of Gravity in Orbital Motion
! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State
More informationSolution Derivations for Capa #8
Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass
More informationPearson Physics Level 30 Unit VI Forces and Fields: Chapter 10 Solutions
Peason Physics Level 30 Unit VI Foces and Fields: hapte 10 Solutions Student Book page 518 oncept heck 1. It is easie fo ebonite to eove electons fo fu than fo silk.. Ebonite acquies a negative chage when
More informationGraphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.
Gaphs of Equations CHAT PeCalculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such
More informationESCAPE VELOCITY EXAMPLES
ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 14 Intoduction: Pehaps one of the most unusual
More informationQuantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w
1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationTORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION
MISN034 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................
More informationCRRC1 Method #1: Standard Practice for Measuring Solar Reflectance of a Flat, Opaque, and Heterogeneous Surface Using a Portable Solar Reflectometer
CRRC Method #: Standad Pactice fo Measuing Sola Reflectance of a Flat, Opaque, and Heteogeneous Suface Using a Potable Sola Reflectomete Scope This standad pactice coves a technique fo estimating the
More informationMultiple choice questions [60 points]
1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chpte The lectic Field II: Continuous Chge Distibutions Conceptul Poblems [SSM] Figue 7 shows n Lshped object tht hs sides which e equl in length. Positive chge is distibuted unifomly long the length
More information